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Basic Productivity Concepts: Meaning and Measurement Productivity is the relationship between output and one or more of the associated inputs used in the production process, as explained in Chapter 2. In that chapter we warned against various misinterpretations and misuses of productivity measures and gave some guidelines for the appropriate interpretation and use of these measures. This chapter develops the concept of productivity more systematically within the framework of production theory, by means of empirical production functions. It explains how partial productivity and multi-factor productivity may be measured within the framework of the economic accounts. It develops further the concepts of output and inputs, which underlie the productivity relationship. This chapter serves as an introduc- tion to Chapters 5 and 6, which explore the conceptual and statistical problems of measuring output and inputs in greater detail. THE FRAMEWORK OF PRODUCTION THEORY The conceptual framework for the measurement of productivity, including changes in productivity over time and differences in productivity among organizations producing similar output, is found in the theory of production. Production theory is most directly relevant to individual firms or other producing units, such as the establishments of multi-establish- ment firms. It may also be used, with appropriate modifications, to analyze the productivity of industries composed of firms or establishments engaged primarily in producing a specified range of products or for the aggregation 35

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36 REPORT OF THE PANEL of industries that comprise the entire business economy. It is not our purpose to go deeply into production theory in this chapter but to summarize some of its main features in order to explain the rationale for the weighting procedures used to aggregate various types of output and of inputs. At the level of the firm or industry, as distinguished from the entire economy, the volume of output (Q) is a function ~ of the volume of services of the basic factors of production, labor (L) and capital (K), of the intermediate products purchased from other firms (X), and of the level of productive efficiency, which changes over time (t). Thus Q = f (L, K, X, t). Each of these variables except t is, of course, an aggregation of various components. In particular, K includes the services of not only created capital structures, equipment, and inventories-but also of land and other natural resources. The services of labor and capital are called primary inputs; the purchased intermediate goods and services are called secondary inputs. Under conditions of competitive equilibrium in labor and other factor markets, producers, in order to minimize costs, use each input up to the point at which the value of its marginal product (the increase in output associated with the use of an additional unit of the input) is equal to its cost per unit, its price. Similarly, in competitive product markets, output is produced up to the point at which its unit cost, including a normal profit, is equal to its price exclusive of indirect taxes. For this reason, product prices for a base period are usually used as weights for combining quantities of various outputs, and base-period input prices are used as weights for inputs. Even though the assumption of competitive markets is often not true, analysts prefer to use objective market prices as weights rather than to attempt adjustments that inevitably involve subjective judgments. In practice, it is more convenient to weight index numbers of inputs by their base-period percentage shares of total costs rather than by physical units of each type of input by unit compensation of price, which gives the same result. As we discuss below, the precise mathematical form of the production function used by an investigator as a framework for measuring real output and inputs affects the types of weighting schemes used, but the erect of alternative weighting schemes does not seem to be significant in the range of data from which estimates are usually made (see Maddala in this volume). When output and inputs have been measured in constant prices over time, ratios of output to individual input classes can be calculated to obtain

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Basic Productivity Concepts: Meaning and Measurement 37 "partial" productivity measures, or ratios of output to all associated inputs may be calculated to obtain a "multi-factor" productivity measure. Changes in multi-factor productivity measures reflect the net saving in the real costs of production achieved over time, that is, increases in productive efficiency generally, if all inputs are included in the denominator. The primary force behind increases in multi-factor productivity, assuming comparable rates of capacity utilization in the periods being compared, is cost-reducing technological progress. But, as we discuss in Chapter 7, other forces also affect productivity, including economies of scale, changes in the quality of resources, and, under the dynamic disequilibrium conditions that prevail in the real world, inter-industry shifts in resources. Partial productivity measures are useful in showing the savings that have been achieved over time in the use of each input per unit of output. Their changes, however, reflect not only changes in productive efficiency but also factor substitutions that result from changes in relative factor prices. In technical jargon, changes in partial productivity reflect move- ments along production functions as factor proportions are changed as well as shifts in production functions due to technological change. When relative input prices change as a result of changing supply and demand forces in factor markets, managers alter input proportions in order to minimize unit cost given the changed set of relative prices. This may also affect the output mix. When the production accounts of all industries are consolidated, inter- industry purchases and sales of intermediate products cancel out, and the real national product is measured in terms of final products including exports and excluding imports. The final goods and services, which are the ones not resold in the same accounting period, comprise private and public consumption and investment, which contribute directly to the economic welfare of people now and in the future. Thus, at the national level the production function is simplified. Final output (Q.) is the product of the services of primary factor inputs alone, and the level of productivity in any given time period (t) is Qn=f'~(L, K, t). The primary factor inputs include those used in making the intermediate products incorporated in the final products or added to inventories as well as those directly used in final production. Real national product can be broken down into what is called the "real product originating" in each industry. This involves subtracting real purchases of intermediate products of each industry from the real value of

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38 REPORT OF THE PANEL its gross production and net inventory change to obtain the real value added Ad. Since intermediate products are now out of the industry output measure, they are also excluded from the inputs, so the product-originat- ing production function for industries (~) becomes analogous to that for the total business economy: V= g(K, L, t). Productivity estimates based on this version of the production function indicate changes in the efficiency with which the primary inputs of an industry are used to add real value to the intermediate products purchased from other industries. By this approach, productivity change in the total economy is a weighted average of estimates of the changes in industry productivity, with the industry proportions of total product used as weights totaling 100 percent. The productivity estimates for industries are therefore directly comparable with the productivity estimates for the economy. Despite the formal consistency of industry and national productivity estimates based on the real product-originating (value-added) approach, there are persuasive reasons for developing estimates of industry produc- tivity relating industry output to all associated inputs (see Gollop and Jorgenson 1980~. First, the exclusion of intermediate inputs from industry measures restricts the generality of the basic model of production, since in principle it is possible to substitute a primary input for a secondary one and vice versa. Second, productivity measures based on total output and inputs explicitly indicate the savings achieved over time in intermediate products, including energy per unit of output as well as the savings in primary factor inputs. It is true that the net productivity measures based on value added also reflect such savings, in that real value added per unit of primary input rises when unit requirements for intermediate inputs are reduced, but the eject is not explicit. In view of the increasing problems of supply of energy and some other materials, it is now more important than ever to quantify the relationships of these inputs to output.) One of the problems with the productivity measures for industries based on total output had been their apparent inconsistency with the economy- wide measures. That is, an industry's total-output productivity rises less than value-added productivity, since for the latter measure intermediate inputs are subtracted from both output and input. The difference in productivity increase shown by the two measures is proportional to the share of purchases of intermediate inputs in the industry's total factor costs. Nevertheless, the productivity measures based on total output can be

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Basic Productivity Concepts: Meaning and Measurement 39 reconciled with the economy-wide measures in which inter-industry sales and purchases of intermediate products cancel out. It has been shown (Gollop in this volume) that the economy-wide measures of productivity growth equal a weighted sum of the industry rates of productivity growth, with weights equal to the ratio of industry output to the aggregate output sold to the final demand categories (the weights total more than 100 percent by the ratio of intermediate costs to final product). The fact that the gross productivity measures for industries can be reconciled with the economy-wide measures enhances the attractiveness of the generalized gross productivity measures for industries. MEASURES OF PRODUCTIVITY We next consider the partial productivity measures usually obtained as ratios of gross or net output to individual classes of inputs, particularly labor. Since multi-factor productivity measures may be obtained as a weighted average of the various partial productivity ratios as well as by other methods, it seems logical to consider the partial measures first. MEASURES OF PARTIAL PRODUCTIVITY Historically, partial productivity measures, particularly ratios of output to the associated labor inputs, were the first type of productivity measures to be developed. Beginning in the 1880s, occasional studies of output per unit of labor input were prepared in the Bureau of Labor and its successor agency, the Bureau of Labor Statistics (BLS). In the 1930s, extensive studies of labor productivity were undertaken by the National Research Project of the Works Progress Administration and by the National Bureau of Economic Research. In 1940 a separate division was created in the BES to prepare regular estimates of output per hour in various industries (and later sectors) of the U.S. economy. That effort has continued to the present day, and current government estimates of productivity are still confined to measures of output per labor hour (except for estimates of multi-factor productivity in farming, which are prepared by the U.S. Department of Agriculture). Most work on multi-factor productivity has been done by private investigators in universities and research institutions, beginning in the 1940s. Since labor is only one of the factor inputs, changes in the output/labor ratio (Q/L), as we noted above, are affected by changes of input mix (factor substitutions) as well as by changes in productive efficiency as measured by multi-factor productivity estimates. If total input is designat- ed by I, then for labor productivity,

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40 REPORT OF THE PANEL Q/L = Q/I X I/L. Rates of change in output per unit of labor input reflect rates of change in both multi-factor productivity and total factor input per unit of labor input. In the U.S. economy and in almost all of its industry divisions, non- labor factor inputs have risen significantly faster than labor inputs. Output (real product) per unit of labor input has therefore generally risen faster than multi-factor productivity, and the output/capital ratio has risen less rapidly (see Figure 3-1~. Since total tangible factor input is a weighted average of the labor and capital inputs, the difference in rates of change between the labor and multi-factor input measures represents the rate of substitution of capital for labor. The next point to make is that the meaning of changes in the output/labor ratio depends on how each term of the ratio is defined and measured. We assume first that labor input is measured as undifferentiated loo J in: cn O 75 At: ? Multi-factor productivity -Real product perunit of capital - Real product per unit of labor 1 1 1 1 1 1 1950 1955 1960 1 965 1 970 YEAR FIGURE 3-1 Companson of indexes of partial and multi-factor productivity for the private domestic economy, 1947-1967 (1958 = 100). (Derived from data points in Kendrick, John W. [1973] Postwar Productivity Trends in the United States, 1948-1969. Table A-l9. Pp. 243-244. New York: Columbia University Press. Used by permission of the National Bureau of Economic Research.)

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Basic Productivity Concepts: Meaning and Measurement 41 hours worked and that quantities of outputs are combined (weighted) by unit value added (factor costs) or by prices. In this case the change in output per hour, measured for an aggregate of products and industries, reflects relative shifts of production among products with differing ratios of factor costs or prices to hours in the base period as well as productivity changes in producing each of these products. Similarly, if there are relative shifts of output to less capital-intensive industries (and thus more value added relative to capital input), the aggregate output/capital ratio increases even though capital requirements per unit of output remain the same in all component industries. If production shifts toward industries with lower unit requirements for energy or for intermediate products as a whole (as in the service industries), the eject of this shift increases the corresponding partial productivity ratio. The general method used to eliminate the eject of shifts of production among industries or products on partial productivity ratios is to weight the component output measures by their base-period requirements for the input to which output is being related. Where possible, BES in its program for measuring productivity in detailed industries weights outputs by unit labor requirements. By this procedure, output per hour measures for an aggregate are unaffected by relative shifts of hours worked among product lines or industries (see Chapter 4~. The formula underlying the industry output indexes of the Bureau of Labor Statistics (1976, chapter 31) follows: if qO and qua represent the base- period and current-period quantities of a given product, lo represents the employee-hours required to produce the product in the base period, and indicates a summation over all types of products: output index = all SI,,q,, A major reason the BES uses unit-employee-hour weights is its interest in changes in unit labor requirements. For this purpose, it is often useful to look at the reciprocal of the output-per-hour measure, that is, employee- hours per unit of output the ratio for two periods of the total hours required to produce a given aggregate.2 Although the BES can usually obtain unit labor requirement weights for product groups and industries, in some industries it has to use substitute weights (usually unit values) for individual products. A change can therefore occur in such industry indexes of output per hour because of changes in output mix, without any change in output per hour for any individual product. This would not be possible for aggregates of industry

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42 REPORT OF THE PANEL productivity indexes, however, with respect to shifts in output among the component industries. Output can, of course, be related to capital input as well as to labor input, and to the components of each broad category for which there are data, yielding a broad spectrum of partial productivity ratios. Output can be related to hours of production and nonproduction workers by occupational groupings and separately to the services of real stocks of structures, equipment, inventories, or land. Industry output measures can also be related not only to primary factor inputs but also to inte~ediate product inputs as a whole and by type, such as energy, raw materials, supplies, and other purchased goods and services. As stressed in Chapter 2, these partial productivity measures are not to be interpreted as measures of changes in efficiency of the particular resource contained in the denominator, but rather as reflections of changes in the use of the resource per unit of output. This may be indicated more clearly when the productivity ratio is inverted. Like measures of output per labor-hour, non-labor productivity ratios also reflect general changes in productive efficiency, factor substitutions, and the weighting procedures used for the output and input components. MEASURES OF MULTI-FACTOR PRODUCTIVITY The development of measures of multi-factor productivity, which relate output to capital as well as to labor inputs, has lagged behind the estimation of output/labor ratios in the United States for several reasons. First, the estimation of production and of employment and hours, based on data available from economic censuses and surveys since the mid- nineteenth century, antedated the estimation of real capital stocks. Estimation of the real capital or "wealth" of the U.S. economy by major sector and industry began on a systematic basis only after World War II (Kuznets 1946, 1961; Goldsmith 1962, 1964~. More fundamentally, the development of national economic accounting systems, which match the national product by sector and industry against the associated labor and non-labor factor costs and other charges against product, did not begin until about 1940. The relationship of national product to factor costs, when expressed in constant product prices and constant factor prices, respectively, provides a measure of changes in the efficiency with which resources are used in production.3 Finally, in the 1950s there was an upsurge of interest in the development of production function theory and its application to the empirical analysis of multi-factor productivity. Statistical production functions for manufac- turing had first been developed by Charles W. Cobb and Paul H. Douglas

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Basic Productivity Concepts: Meaning and Measurement 43 in 1927, but they had not been used to derive productivity estimates. In the 1950s, the new estimates of output and of factor inputs were used within the framework of the Cobb-Douglas production function and other forms of production functions to yield estimates of multi-factor productivity in the U.S. private economy. Early estimates of multi-factor productivity in the United States were prepared by Ian Tinbergen in 1942, by George Stigler in 1947 for a manufacturing industry, by Barton and Cooper in 1948 for agriculture, and in the 1950s by several investigators for the U.S. private domestic economy.4 The basic concept of multi-factor productivity underlying this work is simple: multi-factor productivity = 0~+Q bK In this formulation the capital letters denote index numbers: Q is the real product of the sector (in eject, a price-weighted quantity aggregate); L is labor input, measured as labor-hours in component industries weighted by base-period average hourly labor compensation; K is capital input, assumed to change proportionately to real capital stocks in the various industries, weighted by base-period rates of return; and a and b are the percentage shares of labor and of capital (including land) in the factor income originating in the sector. Other aspects of estimating procedures are discussed in the sections below on output and inputs. Here it is important to stress that multi-factor productivity is measured within an economic accounting framework as a ratio of real product to the associated real factor costs, labor and non-labor. The weights are changed periodically to redect changes in the structure of production and in the relative prices of outputs and of labor and capital inputs. This concept is often called "total factor productivity," but some observers have suggested that this designation may be misleading because intangible inputs resulting from research and development, education, and training are not included with the tangible factor inputs, nor are governmental services to business included when the measures are confined to the private economy.5 At about the same time that multi-factor productivity was being measured using an economic accounts framework, Solow (1957) and other economists took a parametric approach to estimating multi-factor produc- tivity, initially emphasizing the Cobb-Douglas production function: Q = TL aK$.

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44 REPORT OF THE PANEL In this formulation the exponents a and ,8 measure the elasticities of output with respect to labor and capital input or, when index numbers are used, the income shares of the factors. Analysts usually set ,8 equal to 1-a, which imposes the assumption of constant returns to scale.6 Multi- factor input is a geometric weighted mean of the individual inputs, which allows for diminishing returns to the factor that is increasing relatively; see Maddala (in this volume) on weights and the form of the production function. In cross-sectional analysis, the scalar T measures the difference in multi- factor productivity between producing units; in time-series analysis it measures the change in multi-factor productivity over time. In regression analysis of time series of outputs and inputs, (1 + r)t may be used to indicate the average secular rate of growth in the productivity scalar, T (Fabricant 1968~. The rate of change in T may also be obtained for time periods of varying lengths as the difference between rates of change in output and the weighted rates of change in the inputs: AT I AL !\K -= Q -a L -/3 K In this formulation, multi-factor productivity is often called the residual since it reflects all forces, including errors and omissions, other than the measured inputs that contribute to the growth of output. Following Solow's initial work, there has been considerable further development of the production function approach using both the Cobb- Douglas function and other functions involving different concepts of the production process.7 The Cobb-Douglas formulation assumes competitive markets, constant returns to scale, neutral technological change, and constant coeffcients equal to unity, which mean constant shares of the factors in national income. Other formulations allow for economies of scale, variable elasticities of substitution between the factors, and "biased technological change," which means that changing technology may increase the demand for one factor relative to another. Under the latter assumptions, the shares of the factors in national income (factor cost) may be changing over time. This implies either a constant elasticity of substitution that diners from unity or a variable elasticity of substitution. Both types of production functions have been used for empirical investigations. Jorgenson and his associates in studies of productivity prefer to use a translog type of production function, which is consistent with Divisia index numbers.8 By this method, average price weights for successive pairs of years are applied to rates of change in output and in

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Basic Productivity Concepts: Meaning and Measurement 45 input to obtain weighted rates of change in aggregates, which are then linked for successive pairs of years to form continuous index numbers of output, input, and the productivity ratios. The form of the production function is relevant largely in relation to the weights by which the inputs (and outputs) are combined to obtain aggregates and to the interpretation of the results. As a practical matter, since the shares of the factors have not changed drastically over intermediate time periods, different weighting schemes give similar statistical results, as demonstrated in Maddala (in this volume). However, when the weights of outputs and of inputs are not changed periodically, aggregates obtained using relative price weights from an early period show a larger increase than those obtained using weights from a more recent period. This results from the tendency in a dynamic economy for relative changes in quantities (whether of outputs or of inputs) to be negatively correlated with relative changes in their prices, since purchasers tend to substitute goods or services that are becoming relatively cheaper for those that are becoming relatively dearer. The goods or services whose output has grown most tend to have lower relative price weights in a recent period, so their aggregate growth is less than the growth of an aggregate that uses the higher relative price weights of an earlier period for the randily growing outputs. Since this phenomenon applies both to aggregates of output and of inputs, the aggregation bias affects productivity ratios less than it affects the separate variables. Nevertheless, the bias suggests that a system of periodically changing weights is preferable to the use of one set of fixed weights over a long period. Investigators at the National Bureau of Economic Research in studies of production and productivity trends have generally changed average weights every 5 or 10 years in order to reflect possible significant changes in the structure of production and the relative prices of outputs and inputs. Estimates for successive subperiods are then linked so that the index numbers are continuous. OUTPUT National product is measured in terms of final products, which the U.S. Department of Commerce defines as those not resold during a single accounting period. Intermediate products, those purchased by one firm from another for further processing, are not added to final products in order to avoid double counting, since the cost of the intermediate products is already part of the value of final goods and services. Viewed differently, the national product represents a consolidated production statement for all the producing units of the economy, in which inter-industry or inter-firm purchases and sales cancel out. The value of final product equals the cost

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46 REPORT OF THE PANEL of the services of the basic factors of production including profits-plus indirect business taxes minus subsidies, which gives a total value at market price rather than at factor cost. To measure productivity, real product must be related to real factor costs, or inputs. The Department of Commerce confines estimates of the GNP to market transactions, plus several non-market activities with significant market counterparts that serve as a basis for value imputations. The imputations comprise less than 8 percent of the GNP. It is important for productivity measurement that imputations be limited in scope, since independent output and input data are generally not available for non-market activities. National product is also the sum of value added, or product originating in the component sectors, industries, and producing units of the economy. In order for product originating in these components to add to national product, the costs of intermediate products must be deducted from the value of gross production (sales plus the value of changes in inventories of finished goods and goods in process): product originating in a sector, industry, or producing unit, then, equals factor cost plus indirect business taxes minus subsidies. For a component sector as for the whole economy, productivity may be measured as a relationship of real product to real factor costs or inputs. However, as noted earlier, for some purposes it is desirable to relate the total output of industries (or firms) to both factor inputs and inputs of intermediate products in order to show explicitly the savings achieved over time per unit of output in the use of both intermediate products and factors. At all levels of the economy there is controversy about whether to use real gross or net product (i.e., estimates gross or net of capital consumption allowances), since both are net of intermediate products (see discussion in Chapter 5~. The controversy raises the question of whether gross national product should be measured in terms of market prices or unit factor costs, the differences being indirect business taxes minus subsidies per unit of output. It is generally believed that market price weights are appropriate for welfare comparisons but that unit factor cost weights are more appropriate for production and productivity compari- sons since they indicate the relative importance of products in terms of resource inputs and costs (Hicks 1940~. As a practical matter, however, it is a laborious and uncertain process to adjust market prices of all output so as to remove the erects of indirect business taxes minus subsidies. Sensitivity analyses performed on the real GNP estimates for both Canada and the United States by their statistical agencies indicate that such adjustments have little erect on changes in aggregate measures. In order to convert current-value GNP to constant prices, it is theoretically equivalent to weight quantities of outputs by base-period

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Basic Productivity Concepts: Meaning and Measurement . 47 prices or to deflate (divide) values by consistent price indexes. This is also true for the sector, industry, or producing unit estimates of output. The choice should therefore be made on the basis of relative availability of quantity data, or of value and associated price time series. At the economy and sector levels, the deflation approach is generally better, although for many industries and firms, weighted quantity aggregates can be prepared more readily. In some cases, a combination of both methods will yield the most accurate results. For a comparison of the two methods in estimating output of manufacturing industries, see Popkin (in this volume). When quantity data are used, they should relate to the finest possible categories of products. Price deflators should be composed of indexes based on price data for a collection of strictly specified outputs within each product class. The use of fine categories is necessary to ensure that relative shifts in production among differing qualities of products will show up as changes in real output. Even when that is done, however, there are additional problems involved in adjusting data on quantities or prices of specific products for changes in quality through time, often associated with model changes. Related to this are the problems of new or disappearing products, non-standard and custom-made products, and non-market output. There are also broader issues involved in adjusting GNP to provide a better basis for appraising changes in economic welfare. These problems are discussed further in Chapter 5 and in Moss (in this volume) and Scott (in this volume). INPUTS This section is confined to a discussion of the inputs of the basic factors of production. When intermediate products are treated as inputs and related to gross output, the measures are the same as for the intermediate portions of output, and the observations in the preceding section apply. In other words, the output of certain firms (industries) is sold to others for further processing or adding of value prior to final sale, and no new conceptual or measurement problems are introduced. The basic factors of production, however, are stocks, and their inputs are the services rendered by these resource stocks in the production processes during successive time periods. In operational terms, the services may be viewed as time rates of use of the real stocks, with the labor-hours and real capital hours weighted by their base-period prices (average hourly compensation), which presumably reflect their relative marginal produc- tivities in the base period. An increase in real product over a given period relative to real factor cost indicates an increase in factor productivity, since the contributions of the factor services were held constant at the base

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48 REPORT OF THE PANEL period levels. To state the matter differently, the growth in real product over the given period indicates how much factor requirements would have increased with no productivity change, and the change in the ratio of real product to actual factor inputs provides a measure of the productivity growth that actually occurred. The specifications of the input units, like those of the output units, affect the changes in them and thus the estimated change in productivity. There are two major dimensional aspects. The first concerns the detail in which the inputs are defined and measured. If inputs are measured and weighted by detailed industry groupings and if the average base compensation for equivalent inputs differs by industry, then changes in input proportions affect the measured changes in aggregate input but do not affect the measured changes in productivity. But if input is measured in the aggregate without industry breakdown, changes in the industrial mix of the inputs affect the measured change in productivity. In the case of labor, hours worked may also be measured and weighted by occupation, age, sex, or other characteristics associated with differences in pay rates. Changes in mix with respect to these characteristics then affect the aggregate input measure rather than productivity.9 Another dimension of input measurement is the change within particular industries or sectors in the quality of labor and other resource inputs for which rates of compensation differ. The average quality of land and other natural resource capital used in production obviously changes over time. The quality of structures and equipment also changes as a result of technological progress. The quality of labor resources changes primarily because of changes in average education, training, experience, and, possibly, health and vitality. In general, productivity specialists have not adjusted factor inputs for changes in the quality of the inputs within industries and sectors. To some extent, the adjustments for increased average education made by Denison may have this effect and may also reflect changes in the mix of categories of labor with different pay rates due to differential educational attainment. His adjustment of labor-hours to reflect increased vitality as a result of reductions in the average length of the workweek or workyear is also a kind of quality adjustment. It is not important whether inputs are adjusted for quality changes as a result of changes in mix or changes in average quality within various categories or whether quality changes show up instead in the productivity residual. It is important that changes in quality be quantified so that their contributions to changes in output can be estimated, as part of changes either in input or in productivity. This underscores the need for

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Basic Productivity Concepts: Meaning and Measurement 49 investigators to make their concepts and methodologies explicit so that their results may be interpreted correctly. Vanous other specific input dimensions remain to be discussed. Should labor input be measured in terms of hours paid for or hours at the workplace? Should changes in rates of capital utilization be reflected in the input measures? Should real capital stocks be weighted by base-per~od rates of return or should capital compensation be deflated by indexes of the rental price of capital? These and other specific issues in estimating inputs are treated in Chapter 6. NOTES 1. Gollop and Jorgenson (1980) point out that changes in energy consumption are of special importance because in many production processes they are closely related to the rate of capacity utilization. 2. This measure may be obtained directly or as the quotient of indexes of employee-hours and of output: unit-employee-hours index = ~'Q~ . >~ Q = Note that the current-period weighted (Paasche) unit-employee-hours index uses a base-period weighted (Laspeyres) output index divided into the employee-hours index. Conversely, a base-period weighted unit-employee-hours index would be consistent with an output index utilizing current-period weights (see Bureau of Labor Statistics 1976, chapter 31~. 3. This observation was first made by Copeland (1937, p. 31~. 4. Writers in the 1950s included Schmookler, Kendrick, Abramovitz, and Fabricant; see the summary volume by Kendrick (1961a). 5. See "Director's Comment" by Stanley Ruttenberg in Kendrick (1961a). 6. This was originally done because early estimates without this constraint gave values of a and l] whose sum did not differ significantly from unity. However, this may be an artifact arising from the estimation of the share of profits as a residual. 7. For a review of the literature to 1970, see Nadiri (1970~. 8. If one replaces the translog production function by a homogeneous quadratic function, the appropriate indexes are Fisher's ideal index numbers. This result, due to Diewert (1971), can be found in Maddala (in this volume). 9. Kendrick, Jorgenson, and others have weighted inputs by industry groups in greater or lesser detail and have interpreted their productivity ratios accordingly. Denison has not used internal industry weights and therefore has included inter- industry shifts as one of the explanations of changes in the productivity residual. Denison and Jorgenson have adjusted labor input for changes in composition with respect to age, sex, and certain other characteristics; Kendrick has not, so that his productivity estimates have been affected by such shifts.